a post with math

In this post I’ll talk about the shell models of turbulence. The shell models are a chaotic dynamical system which is able to reproduce some of the statistical properties of turbulence in the hydrodynamic limit, such was intermittency and anomalous scaling exponents. It can be though as simplified view of the cascade of energy in 3D turbulence.

The reason why people use these models is because they are much cheaper than simulating 3D turbulence! 3D turbulence scales exponentially with the Reynolds number as $Re^{9/4}$! The Reynolds number ($Re = U l / \nu$) is a non-dimensional number that relates a characteristic length scale $l$, velocity scale $v$ and viscosity $\nu$. It represents the ratio between inertial and viscous forces. We are interested in cases where $Re \approx \infty$.